Color a \(d\) -set \(T\subset N(x)\)  green if \(N(T)=\lbrace x\rbrace \) , blue if \(1<|N(T)|<r\)  and red if \(|N(T)|\ge r\) . Note that if \(y\in B\setminus \lbrace x\rbrace \) , then any \(d\) -set \(T\subset N(x)\cap N(y)\)  is colored blue or red. If there exists some \(R\subset N(x)\)  of size \(r\)  such that every \(T\subset R\)  of size \(d\)  is red, then condition 1. holds. However, if \(\ell \)  is sufficiently large, by Ramsey's theorem (see [1]} or [2]}) we know that for every set \(L\subset N(x)\cap N(y)\)  of size \(\ell \) , there exists a subset \(R\subset L\)  of size \(r\)  such that all \(d\) -sets in \(R\)  have the same color.
